📐 Lines & Angles
🔺 Triangles
⭕ Circles
📏 Mensuration & 3D
Geometry — Shapes, Spaces & Theorems
Geometry is the study of shapes and their properties. From the pyramids of Egypt to modern architecture, geometry describes the physical world. Every competitive exam tests geometry — SSC CGL has 8-10 questions, Banking 5-6, CDS/NDA 10-12, and CTET 8-10.
Part 1
📐 Lines & Angles
Where It All Begins
The Egyptian Rope-Stretchers
Every year the Nile flooded, washing away farmland boundaries. The Egyptians needed to re-mark fields. They used stretched ropes — straight lines — to measure and divide land. They discovered that a rope with knots at 3, 4, and 5 units made a perfect right angle. That was geometry's birth: measuring the earth (geo = earth, metry = measure).
Geometry started as a practical tool: dividing land, building structures, measuring distances.
Angle Basics
An angle is formed when two rays meet at a point. Measured in degrees (°), a full circle is 360°.
Acute
0° < θ < 90°
Right
θ = 90°
Obtuse
90° < θ < 180°
Straight
θ = 180°
Reflex
180° < θ < 360°
Complementary
Sum = 90°
Supplementary
Sum = 180°
Vertically Opposite
Equal
Parallel Lines & Transversals
When a transversal cuts two parallel lines, specific angle pairs are equal:
- Corresponding angles (F-pattern): equal
- Alternate interior angles (Z-pattern): equal
- Co-interior angles (C-pattern): supplementary (sum = 180°)
Corresponding angles are equal when a transversal cuts parallel lines.
Angle Sum Property
Sum of angles in a triangle = 180°
Sum of angles in a quadrilateral = 360°
Sum of exterior angles (any polygon) = 360°
Sum of angles in a quadrilateral = 360°
Sum of exterior angles (any polygon) = 360°
1
Two complementary angles differ by 20°. Find the larger angle.
▶
Let angles be x and y with x + y = 90° and x − y = 20°.
Adding: 2x = 110°, so x = 55°. Then y = 90° − 55° = 35°.
Larger angle = 55°.
Adding: 2x = 110°, so x = 55°. Then y = 90° − 55° = 35°.
Larger angle = 55°.
Part 2
🔺 Triangles — The Strongest Shape
Nature's Favorite Shape
Why Bridges Use Triangles
A square frame wobbles. Add a diagonal — two triangles — and it's rigid. Triangles are the only polygon that cannot change shape without changing side lengths. That's why bridges, cranes, and roof trusses use triangles. A triangle is the simplest rigid structure in engineering.
Triangles are uniquely rigid. That's why they're everywhere in construction.
Types of Triangles (by sides)
- Equilateral: all sides equal, all angles = 60°
- Isosceles: two sides equal, base angles equal
- Scalene: all sides different
Types (by angles)
- Acute: all angles < 90°
- Right: one angle = 90° (hypotenuse opposite right angle)
- Obtuse: one angle > 90°
Congruence (Same Size & Shape)
SSS
Three sides equal
SAS
Two sides + included ∠
ASA
Two angles + included side
AAS
Two angles + non-included side
RHS
Right + Hypotenuse + Side
Similarity (Same Shape, Any Size)
Two triangles are similar if corresponding angles are equal and sides are in proportion.
Similarity Shortcuts
AA (two angles equal) ⇒ triangles are similar
No need to check sides! If angles match, similarity follows automatically.
Pythagoras Theorem
In a right triangle: a² + b² = c², where c is the hypotenuse (longest side).
Pythagoras: a² + b² = c². Works only for right triangles.
Pythagorean Triplet
(3,4,5), (5,12,13), (7,24,25), (8,15,17)
Common Ratios
30-60-90: 1:√3:2
45-45-90: 1:1:√2
45-45-90: 1:1:√2
2
Find the height of an equilateral triangle of side 12 cm.
▶
In an equilateral triangle, the altitude splits it into two 30-60-90 right triangles.
Half the base = 6 cm. Using the 30-60-90 ratio 1:√3:2:
Short leg (half-base) = 6, long leg (height) = 6√3.
Height = 6√3 cm.
General formula: Height of equilateral triangle = (√3/2) × side.
Half the base = 6 cm. Using the 30-60-90 ratio 1:√3:2:
Short leg (half-base) = 6, long leg (height) = 6√3.
Height = 6√3 cm.
General formula: Height of equilateral triangle = (√3/2) × side.
Key Centers of a Triangle
- Centroid (G): intersection of medians — divides each median in ratio 2:1
- Incenter (I): intersection of angle bisectors — center of incircle
- Circumcenter (O): intersection of perpendicular bisectors — center of circumcircle
- Orthocenter (H): intersection of altitudes
Part 3
⭕ Circles — The Perfect Shape
Wheels & Cycles
Why Circles Are Everywhere
A circle is the set of all points at a fixed distance (radius) from a center. The wheel — humanity's greatest invention — works because every point on a circle's rim is the same distance from the axle. That gives smooth rotation. The Babylonians divided the circle into 360° because it's divisible by 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 45, 60, 72, 90, 120, and 180 — the most divisible number possible.
The circle's symmetry and the divisibility of 360 make it the natural unit for angles.
Circle Theorems — The Exam Shortcuts
Left: All radii equal. Right: Angle in a semicircle is always 90°.
Tangent-Radius
Tangent ⟂ radius at point of contact
Two Tangents
From external point, tangents are equal
Angle in Semicircle
Angle in semicircle = 90°
Equal Chords
Equal chords subtend equal angles at center
Cyclic Quadrilateral
Opposite angles sum to 180°
Alternate Segment
Angle between tangent and chord = angle in alternate segment
Tangent Length Formula
Length of tangent from external point P to circle center O: PT² = OP² − r²
3
Two circles of radii 8 cm and 6 cm touch externally. Find distance between their centers.
▶
When circles touch externally, distance between centers = sum of radii.
Distance = 8 + 6 = 14 cm.
For internal tangency: distance = difference of radii = |8 − 6| = 2 cm.
Distance = 8 + 6 = 14 cm.
For internal tangency: distance = difference of radii = |8 − 6| = 2 cm.
Part 4
📏 Mensuration & 3D Geometry
Measuring Everything
The King's Dilemma
A king ordered a gold crown. He gave the goldsmith pure gold. The crown came back the right weight, but was it pure gold? Archimedes realized: submerge the crown in water. The volume of water displaced equals the crown's volume. Compare density = mass/volume with pure gold's density. He ran naked through the streets shouting "Eureka!" — Greek for "I found it!" That was the birth of mensuration: measuring volume by displacement.
Density = mass/volume. Volume of irregular objects = water displaced.
Perimeter & Area (2D Shapes)
Square
A = a², P = 4a
Rectangle
A = l × b, P = 2(l+b)
Triangle
A = ½ × b × h
Circle
A = πr², C = 2πr
Trapezium
A = ½(a+b) × h
Parallelogram
A = b × h
Rhombus
A = ½ × d₁ × d₂
Sector
A = (θ/360) × πr²
Heron's Formula
Area of triangle with sides a, b, c: A = √[s(s−a)(s−b)(s−c)] where s = (a+b+c)/2
Useful when you know three sides but not the height.
Surface Area & Volume (3D Shapes)
Cube
V = a³, SA = 6a²
Cuboid
V = l × b × h, SA = 2(lb+bh+hl)
Cylinder
V = πr²h, CSA = 2πrh
Cone
V = ⅓πr²h, CSA = πrl
Sphere
V = ⁴⁄₃πr³, SA = 4πr²
Hemisphere
V = ⅔πr³, CSA = 2πr²
Note: CSA = Curved Surface Area. For a cone, l = slant height = √(r² + h²).
Volume Combination Problem
A sphere of radius 6 cm is melted and recast into a cylinder of height 8 cm. Find the cylinder's radius.
Volume stays same in melting and recasting.
Sphere volume = ⁴⁄₃π(6)³ = 288π cm³.
Cylinder volume = πr² × 8 = 8πr².
Equate: 8πr² = 288π → r² = 36 → r = 6 cm.
Volume stays same in melting and recasting.
Sphere volume = ⁴⁄₃π(6)³ = 288π cm³.
Cylinder volume = πr² × 8 = 8πr².
Equate: 8πr² = 288π → r² = 36 → r = 6 cm.
4
Find the area of a triangle with sides 13 cm, 14 cm, 15 cm.
▶
Semi-perimeter s = (13 + 14 + 15)/2 = 42/2 = 21.
Heron's formula: A = √[21(21−13)(21−14)(21−15)]
= √[21 × 8 × 7 × 6]
= √(21 × 336) = √7056 = 84 cm².
This is a well-known triple — area is always 84 for a 13-14-15 triangle.
Heron's formula: A = √[21(21−13)(21−14)(21−15)]
= √[21 × 8 × 7 × 6]
= √(21 × 336) = √7056 = 84 cm².
This is a well-known triple — area is always 84 for a 13-14-15 triangle.
🎯 Practice — 12 Questions
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